General exact solutions of certain second-order homogeneous algebraic differential equations

Abstract

The general exact solutions of the second-order homogeneous algebraic differential equationis studied, where M, C and K are three known bounded linear operators on a Banach space X . If λ ∈ C such that λ 2 M +λC + K is invertible, an anti-triangular operator matrix E λ is defined. When E λ is Drazin invertible, a necessary and sufficient condition is obtained in terms of E λ and its Drazin inverse E D λ under which q(t) ∈ C 2 (R, X ) becomes a solution of (0.1). Specifically, a formula for E D λ is derived under the assumptions that X is a Hilbert space, M, C and K are positive semi-definite operators on X with the summation M + C + K being positive definite.